Consider an infinite sequence of times \(t = t_1,t_2,...\)  and assume that the “if” condition is true for any time-step in this sequence. Since \(\eta _t\)  is a winning strategy for the learning player in the Gale-Stewart game \(\mathcal {G}\) , we have that there exists some finite time \(t^\star \)  so that \(\mathcal {H}_{\xi _1,\eta _1,...,\xi _{t^\star },\eta _{t^\star }}= \emptyset .\)  By the structure of the “if” condition, we have that \(\xi _i = (x_{t_i - \ell _{t_i}-1}+1, ..., x_{t_i})\)  and \(\eta _i = (y_{t_i - \ell _{t_i}-1}+1, ..., y_{t_i})\) . But since the sequence is consistent with \(\mathcal {H}\) , the class \(\mathcal {H}_{\xi _1,\eta _1,...,\xi _{t^\star },\eta _{t^\star }}\)  must contain the hypothesis that makes it consistent which is a contradiction since the class is empty. For the measurability of the strategies, we refer to Remark 5.4 of [1]}.
